Hausdorff dimension of sets with restricted, slowly growing partial quotients in the semi-regular continued fraction

Abstract

We consider sets of irrational numbers whose partial quotients a σ,n in the semi-regular continued fraction expansion obey certain restrictions and growth conditions. Our main result asserts that, for any sequence σ ∈ {−1, 1} N in the expansion, any infinite subset B of N and for any function f on N with values in [min B, ∞) and tending to infinity, the set of irrationals in (0, 1) such that a σ,n ∈ B, a σ,n ≤ f (n) for all n ∈ N and a σ,n → ∞ as n → ∞ is of Hausdorff dimension τ (B)/2, where τ (B) is the exponent of convergence of B. We also prove that for any σ ∈ {−1, 1} N and any B ⊂ N, the set of irrationals in (0, 1) such that a σ,n ∈ B for all n ∈ N and a σ,n → ∞ as n → ∞ is also of Hausdorff dimension τ (B)/2. To prove these results, we construct nonautonomous iterated function systems well-adapted to the given restrictions and growth conditions, and then apply the dimension theory developed by Rempe-Gillen and Urbański.